\documentclass{letter}

\usepackage{amsmath}

\renewcommand{\d}{\mathrm{d}}
\newcommand{\diff}[2]{\frac{\d #1}{\d #2}}
\newcommand{\diffsq}[2]{\frac{\d^2 #1}{\d #2^2}}
\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\pdiffsq}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\renewcommand{\vec}[1]{\boldsymbol{#1} }

\begin{document}

a)

Take $\mu = i$ and sum the remaining indices:

\begin{eqnarray*}
J_i &=& \frac{\sigma}{c} F_{i\mu} u^\nu + \frac{1}{c^2} ( u^\nu J_\nu ) u_i \\
&=& \frac{\sigma}{c} \left( F_{i0} u^0 + F_{ij} u^j \right) + \frac{1}{c^2} \gamma \left( \rho c - \vec u \cdot \vec J \right) u_i \\
&=& \frac{\sigma}{c} \left( (- E_i) (-\gamma c) + (- \epsilon_{ijk} B_k )(-u_j) \right) + \gamma \left( \frac{\rho}{c} - (\vec \beta \cdot \vec J ) \right) \beta_i \\
&=& \frac{\sigma}{c} \gamma \left( c \vec E + (\vec u \times \vec B) \right)_i + \gamma \left( \frac{\rho}{c} - (\vec \beta \cdot \vec J ) \right) \beta_i \\
\Rightarrow \vec J  &=& \frac{\sigma}{c} \gamma \left( c \vec E + (\vec u \times \vec B) \right) + \gamma \left( \frac{\rho}{c} - (\vec \beta \cdot \vec J ) \right) \vec \beta
\end{eqnarray*} 

b)

c)



\end{document}